Theorem 2  [1]}
Suppose that \(A\)  is a real \(m \times n\)  matrix with singular values \(\sigma _{1} \ge \sigma _{2} \ge \sigma _{3} \ge \cdots \) . Choose a target rank \(k \ge 2\)  and an oversampling parameter \(l \ge 2\) , where \(k+l \le \min \lbrace m, n\rbrace  .\)  Then \(\mathtt {RSVD}\)  called on \(W=(AA^*)^q A\)  with target rank \(k\) , and oversampling parameter \(l\)  produces an orthonormal approximate basis \(P\)  which satisfies
\(\begin{aligned}\mathbb {E}\Vert (A-PP^*A)\Vert \le \left[1+\sqrt{\frac{k}{l-1}}+\frac{e \sqrt{k+l}}{l} \cdot \sqrt{\min \lbrace m, n\rbrace -k}\right]^\frac{1}{2q+1} \sigma _{k+1}\end{aligned}\) 
